SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2018-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article |
| _version_ | 1827994935730634752 |
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| author | THOMAS A. HULSE CHAN IEONG KUAN DAVID LOWRY-DUDA ALEXANDER WALKER |
| author_facet | THOMAS A. HULSE CHAN IEONG KUAN DAVID LOWRY-DUDA ALEXANDER WALKER |
| author_sort | THOMAS A. HULSE |
| collection | DOAJ |
| description | The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions $k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem. |
| first_indexed | 2024-04-10T04:47:40Z |
| format | Article |
| id | doaj.art-dd999b252e8f4d03997dfa9e10de1f82 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:40Z |
| publishDate | 2018-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-dd999b252e8f4d03997dfa9e10de1f822023-03-09T12:34:35ZengCambridge University PressForum of Mathematics, Sigma2050-50942018-01-01610.1017/fms.2018.26SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEMTHOMAS A. HULSE0CHAN IEONG KUAN1DAVID LOWRY-DUDA2ALEXANDER WALKER3Mathematics Department, Boston College, Chestnut Hill, MA 02467, USA;School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong Province, 519082, China;Warwick Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK;Department of Mathematics, Rutgers University Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA;The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions $k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article11N37 (primary)11F30 (secondary) |
| spellingShingle | THOMAS A. HULSE CHAN IEONG KUAN DAVID LOWRY-DUDA ALEXANDER WALKER SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM Forum of Mathematics, Sigma 11N37 (primary) 11F30 (secondary) |
| title | SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM |
| title_full | SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM |
| title_fullStr | SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM |
| title_full_unstemmed | SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM |
| title_short | SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM |
| title_sort | second moments in the generalized gauss circle problem |
| topic | 11N37 (primary) 11F30 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article |
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